Weighted unipolar Hardy inequality with optimal constant

Anna Canale, Francesco Pappalardo, Ciro Tarantino

Published 2018-12-07Version 1

\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_\mu\int_{{\R}^N}\frac{\varphi^2 }{|x|^2}\, d\mu\le \int_{{\R}^N} |\nabla\varphi|^2 \, d\mu + K \int_{\R^N}\varphi^2 \, d\mu \quad \forall\, \varphi \in H_\mu^1 %\qquad c\le c_\mu, \end{equation*} in the context of the study of the Kolmogorov operators \begin{equation*} Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u, \end{equation*} with $\mu$ probability density in $\R^N$, and of the related evolution problems. We prove the optimality of the constant $c_\mu$ and state existence and nonexistence results following the Cabr\'e-Martel's approach \cite{CabreMartel} and using results stated in \cite{GGR, CGRT}. \end{abstract}